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Analysis-suitable $G^1$ multi-patch parametrizations for $C^1$ isogeometric spaces

Annabelle Collin, Giancarlo Sangalli, Thomas Takacs

arXiv:1509.07619v2math.NAcs.NA

TL;DR

Optimal approximation by C1 isogeometric spaces on multi-patch geometries is difficult to characterize, and continuity constraints can over-constrain the space. This paper defines analysis-suitable G1 parametrizations and studies the structure and approximation of C1 spaces under h-refinement. Its results associate optimal approximation with analysis-suitable geometries and suboptimal convergence, sometimes locking, with parametrizations outside that class.

  • Problem

    Optimal approximation properties of C1 isogeometric spaces on complex multi-patch geometries remain an open and challenging problem.

    Source: §1 · pp. 1–2

  • Method

    The paper defines analysis-suitable G1 parametrizations and analyzes C1 isogeometric spaces under h-refinement theoretically and numerically.

    Source: §1 · p. 3, §4 · p. 9, §5 · p. 10

  • Results

    Analysis-suitable G1 geometries allow optimal approximation numerically, whereas parametrizations outside the class yield suboptimal order and can prevent convergence through C1 locking.

    Source: §9 · p. 26

  • Takeaways & Limitations

    The approximation behavior of C1 isogeometric spaces depends on the geometry parametrization, with the analysis-suitable G1 class identified as permitting optimal approximation.

    Source: §9 · p. 26

    Numerically computing the C1 continuity conditions is difficult for non-analysis-suitable geometries.

    Source: §8.4 · p. 25

Abstract

from arXiv · show

One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from $p$-degree splines (and extensions, such as NURBS), they enjoy up to $C^{p-1}$ continuity within each patch. However, global continuity beyond $C^0$ on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multi-patch domains that have a parametrization which is only $C^0$ at the patch interface. On such domains we study the $h$-refinement of $C^1$-continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the $C^1$-continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recent studies by Kapl et al. have given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) $C^1$ splines. This is the starting point of our study. We introduce the class of analysis-suitable $G^1$ geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of $C^1$ isogeometric spaces over analysis-suitable $G^1$ parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of $C^1$ isogeometric spaces is prevented.

1. Introduction

Constructing C1 isogeometric spaces with optimal approximation on complex multi-patch geometries remains challenging. This work studies C0-at-the-interface parametrizations, introduces analysis-suitable G1 geometries, and examines how parametrization affects h-refinement.

1. Introduction

  • Optimal approximation by smooth isogeometric spaces on complex multi-patch geometries remains an open challenge, while C1 continuity across patches is not well understood.Source: §1 · pp. 1–2
  • The paper focuses on parametrizations that are only C0 at patch interfaces, rather than those smooth except near extraordinary points.Source: §1 · p. 2, Figure 1 · p. 2
  • Prior numerical tests found optimal convergence for cubic-or-higher C1 splines on bilinear two-patch geometries, but also showed C1 locking on a non-bilinear geometry.Source: §1 · p. 3
  • The paper develops theory for h-refinement across degrees and regularities, defining analysis-suitable G1 parametrizations that include bilinear cases and studying approximation under other parametrizations.Source: §1 · p. 3
  • The study does not provide classical error estimates, and the existence and construction of analysis-suitable parametrizations for arbitrary geometries remain open.Source: §1 · p. 3

2. Planar multi-patch spline parametrizations and isogeometric spaces

The section sets up a uniform spline-patch framework and defines isogeometric functions through patchwise pullbacks. Adjacent patches share a parametrized interface, with assumptions simplifying the configurations under study.

2. Planar multi-patch spline parametrizations and isogeometric spaces

  • The spline spaces use degree p, continuity C^r at interior knots, a uniform mesh of size h, and knot multiplicity p-r.Source: §2 · p. 3
  • The setup excludes hanging nodes and assumes regular partitions, spline patches, and nonsingular patch parametrizations.Source: §2 · pp. 3–4
  • For simplicity, all patches use the same degree and continuity parameters, with uniform knot meshes and matching spline-space conventions.Source: §2 · pp. 3–4
  • Adjacent patches are aligned by linear coordinate transformations and their parametrizations agree along the shared interface.Source: §2 · p. 4
  • An isogeometric function is represented patchwise by composing the physical function with each patch parametrization, with interface smoothness considered through its graph parametrization.Source: §2 · p. 5

3. C1 isogeometric spaces

The section characterizes C1 continuity across an interface through tangent-plane compatibility and the graph's G1 continuity. Geometry-dependent coefficient functions express the parametric continuity condition.

3. C1 isogeometric spaces

  • The interface tangent-plane condition is expressed through geometry-dependent coefficients α(L), α(R), and β, determined up to a common factor by the patch parametrizations.Source: §3 · p. 7
  • For planar domains, coefficients satisfying the geometric interface relation always exist, and the resulting coefficient condition expresses C1 continuity in parametric coordinates.Source: §3 · pp. 6, 8
  • An isogeometric function is globally C1 across an interface if and only if its graph parametrization is G1-continuous there.Source: §3 · pp. 5, 8
  • The same equivalence places C1 isogeometric spaces over C0 planar multi-patch parametrizations within the isoparametric framework.Source: §3 · pp. 8–9

4. Analysis-suitable G1 parametrizations

Analysis-suitable G1 parametrizations restrict the interface coefficient functions to linear polynomials. This class includes bilinear patches and extends to higher-order polynomial parametrizations.

4. Analysis-suitable G1 parametrizations

  • An interface is analysis-suitable G1 when the coefficients α(L), α(R), β(L), and β(R) can be chosen as linear polynomials satisfying the geometric relations.Source: §4 · p. 9
  • Analysis-suitable G1 parametrizations are considered in the isoparametric framework for C1 isogeometric spaces.Source: §4 · p. 9
  • Every pair of bilinear patch parametrizations is analysis-suitable G1, and the class includes higher-order polynomial patches as well.Source: §4 · p. 10

5. Two-patch geometry

For two-patch geometries, C1 approximation is governed by the trace and transversal-derivative spaces induced by the interface and its parametrization. Analysis-suitable G1 geometry yields optimal h-refinement when r < p −1, while higher-degree geometry coefficients can prevent optimal convergence or cause locking.

5. Two-patch geometry

  • The interface analysis uses the trace and directional derivative along a transversal vector d, which depends on the geometry coefficients.The transversal derivative can be a rational function of degree higher than p.Source: §5 · p. 10
  • The two-patch study focuses on a single interface because that configuration isolates the C1 constraints imposed at each patch interface.Source: §5 · p. 10

5.1. AS G1-continuous two-patch geometry

  • For AS G1 two-patch geometries, the trace and transversal-derivative spaces support optimal approximation of V1 when r < p −1.The function trace contains degree-p splines with regularity at least r+1, while the transversal derivative trace contains degree-(p−1) splines with regularity at least r.Source: §5.1 · p. 11
  • When r = p −1 and the parametrization is nontrivial, C1 locking occurs: h-refinement does not improve trace and transversal-derivative approximation independently.In the special trivial G1 configuration, the parametrization can instead be made C1 by scaling one patch's transverse parameter.Source: §5.1 · pp. 11–12
  • Theorem 2 gives h-independent dimensions for G0 if either β coefficient is nonzero, and for G1 if α(L) and α(R) are linearly independent.Source: §5.1 · p. 11

5.2. General two-patch geometry

  • Beyond AS G1, the studied identity-left-patch setting characterizes the C1 space through the degrees of the right-patch α and β coefficients.Theorem 3 considers α(R) of degree pα and distinguishes cases for β(R); the resulting trace-space structure depends on these degrees.Source: §5.2 · p. 13
  • Optimal h-refinement convergence is impossible if deg α(R) > 1 or deg β(R) > 1; C1 locking occurs when either degree is at least p −r.Source: §5.2 · p. 14

6. C1 isogeometric spaces on surfaces

The paper extends its C1 isogeometric framework from planar domains to surface patches, relating C1 continuity of functions to G1 continuity of their graphs. For the specified surface parametrizations, the planar AS G1 approximation results carry over.

6. C1 isogeometric spaces on surfaces

  • The surface extension requires the surface itself to be C1, and the discussion simplifies the setting to two patches with one interface.Source: §6 · p. 14
  • A surface isogeometric function belongs to V1 exactly when its graph is G1-continuous across each interface.Source: §6 · p. 15
  • When the surface parametrization satisfies the stated projection form and AS G1 conditions, the planar results apply directly to the surface spaces.The isogeometric spaces over these G1 and AS G1 multi-patch parametrizations fit in the isoparametric framework.Source: §6 · pp. 15–16

7. NURBS spaces

The paper extends its results to NURBS through rational geometry constructions based on AS G1 surface patches. It also notes that not every multi-patch NURBS geometry fits this framework.

7. NURBS spaces

  • The planar results extend beyond spline spaces, including to NURBS through rational planar geometries generated from AS G1 surface patches.Source: §7 · p. 16
  • Not all multi-patch NURBS geometries fit the framework: their surface patches may fail to be G1 or even C0, although the graph patches join G1.The circle example has a graph surface that is not AS G1.Source: §7 · p. 16

8. Numerical tests

The numerical tests solve a bilaplacian model with an analytic exact solution under h-refinement, examining analysis-suitable and non-analysis-suitable G1 geometries. The AS examples show the predicted convergence behavior, while the non-AS examples exhibit locking or suboptimal convergence.

8.1. Model problem

  • The model problem is a bilaplacian with data chosen to give an analytic exact solution, and homogeneous value and normal-derivative boundary conditions.The unknown is w and the normal is the unit normal to the boundary.Source: §8.1 · p. 17

8.2. Analysis-suitable G1 geometries

  • Across the AS tests, the study uses five mappings with varying patch counts and degrees p = 3, 4, 5, testing regularities 1 ≤ r ≤ p − 1.The numerical tests were obtained with IGATOOLS.Source: §8 · p. 16, §8.2 · p. 17

8.2.1. Two-patch geometry (L-shape)

  • For the L-shaped domain, convergence is optimal if and only if r + 2 ≤ p; for p = 3 and r = 2, C1 locking occurs and the interface solution equals 0.Decreasing regularity circumvents C1 locking.Source: §8.2.1 · p. 18, Figure 5 · p. 18

8.2.2. Multi-patch geometries (triangle, quarter of a circle and rectangle)

  • On the three-patch triangle and quarter-circle AS geometries, p = 3, r = 2 locks, while reducing to r = 1 circumvents locking.The quarter-circle uses NURBS patches obtained from an AS G1 surface in homogeneous coordinates; the expected convergence orders are presented for both geometries.Source: §8.2.2 · p. 19, Figure 8 · p. 19
  • The four-patch rectangle and five-patch smooth-boundary domain show the expected convergence orders; the latter uses bi-quadratic patches with a C1 boundary.Its interior control points are selected to satisfy the AS G1 conditions.Source: §8.2.2 · p. 20, Figure 10 · p. 20, Figure 11 · p. 21

8.3. Non-analysis-suitable G1 geometries

  • The tests compare a distorted two-patch rectangle with an identity-mapped reference and also examine a five-patch circle.The rectangle has a linear left patch and a quadratically distorted right patch; the circle tests rational parametrizations outside the paper's Section 7 framework.Source: §8.3 · p. 20, §8.3.1 · p. 21, §8.3.2 · p. 23

8.3.1. Two-patch geometry (quadratically distorted rectangle)

  • In the distorted rectangle, p = 3 locks for both r = 2 and r = 1, while p = 4, r = 1 converges but is not expected to converge optimally.For the H2 error, the stated order is h3 rather than the optimal order; numerical tests make the suboptimality difficult to measure.Source: §8.3.1 · pp. 21–22

8.3.2. Multi-patch geometry (circle)

  • For the five-patch circle, p = 3 locks at r = 2 and r = 1; p = 4, r = 1 converges, but the reported convergence behavior is suboptimal in all tested cases.The authors note that their theory does not apply to this rational parametrization, whose homogeneous representation is not AS G1.Source: §8.3.2 · p. 23, Figure 17 · p. 24

8.4. Numerical implementation of C1 continuity

The implementation imposes C1 continuity numerically by computing a null space after incorporating boundary and C0 constraints. For non-AS G1 geometries, separating nonzero eigenvalues from numerical zeros can become difficult under refinement.

8.4. Numerical implementation of C1 continuity

  • The method forms a symmetric C1 constraint matrix and uses a null-space basis after applying boundary and C0 constraints.The resulting basis maps the reduced solution back to the initial unconstrained spline basis.Source: §8.4 · p. 24
  • For non-AS G1 geometries, nonzero eigenvalues may not separate sufficiently from numerical zeros, making the C1-constraint null space hard to compute at later refinements.The paper illustrates this issue for the distorted rectangle with p = 4, r = 1.Source: §8.4 · p. 25, Figure 18 · p. 25
  • In the L-shaped comparison, numerical zeros are around 10^-14 and nonzero eigenvalues exceed 10^-2, making them easier to distinguish than in the distorted rectangle.The two configurations have structurally equivalent control-point grid topology.Source: §8.4 · p. 25, Figure 19 · p. 25

9. Conclusions

The paper identifies interface over-constraints in C1 isogeometric spaces as reducing approximation order, and finds numerically that AS G1 parametrizations allow optimal approximation. It also outlines limits of the study and open questions about geometry flexibility and surface constructions.

9. Conclusions

  • AS G1 parametrizations allow optimal approximation numerically, whereas non-AS G1 parametrizations yield suboptimal order and can prevent convergence under h-refinement.Over-constraining the interface trace or transversal derivative reduces approximation order; the worst case is termed C1 locking.Source: §9 · pp. 25–26
  • A complete approximation theory with classical error estimates remains undeveloped and is left for future work.Source: §9 · p. 26
  • The study focuses mainly on planar B-spline geometries, with brief discussion of surfaces and NURBS patches.Numerical tests include bilaplacian problems on several AS and non-AS G1 geometries, and implementing C1 conditions is non-trivial for complex non-AS G1 configurations.Source: §9 · p. 26
  • The flexibility question asks whether patches interpolating given boundary curves can form an AS G1 geometry; bilinear patches solve this for piecewise linear boundaries.A piecewise biquadratic AS G1 parametrization of a C1 simply-connected domain is also presented, while arbitrary degrees and topology remain for further investigation.Source: §9 · p. 26
  • For surfaces, changing patch interiors to achieve AS G1 suitability is not feasible when that changes the surface itself.AS G1 constructions are possible when the surface is given as the image of a planar domain; explicit construction in the general setting remains open.Source: §9 · p. 26

Appendix A: AS G1 parametrization of a quarter of circle

The appendix constructs the three-patch example using quadrilaterals mapped from a reference triangle and a shared triangular Bézier mapping. The resulting patch maps are rational bi-quadratic, allowing the paper’s theory to apply to the example.

Appendix A: AS G1 parametrization of a quarter of circle

  • Each patch is built by combining a bilinear map from the reference triangle to a quadrilateral with a global map to the surface patch in homogeneous coordinates.The construction follows the NURBS setting and ideas in [8] and [21, Section 3.4].Source: Appendix A · p. 27
  • The quadrilaterals use one triangle corner, two adjacent edge midpoints, and the centroid; a degree-2 triangular Bézier map supplies the global mapping.Each bilinear map is determined by its quadrilateral image, up to rotations of the parameter domain.Source: Appendix A · p. 27
  • The resulting component maps are rational bi-quadratic, so the theory from Sections 6 and 7 applies to the Figure 7 example.Figure 20 shows the mappings and corresponding domains.Source: Appendix A · p. 27, Figure 20 · p. 27

Appendix B: AS G1 parametrization of a smooth simply-connected domain

The domain comprises five bi-quadratic patches with a C1-smooth boundary. Its central patch is a shifted unit square, and the left, bottom, and right patch parametrizations are rotations of the top patch.

Appendix B: AS G1 parametrization of a smooth simply-connected domain

  • The domain consists of five bi-quadratic patches and has a C1-smooth boundary.Source: Appendix B · p. 28
  • The central patch is the shifted unit square [-1/2, 1/2]^2.Source: Appendix B · p. 28
  • The left, bottom, and right patch parametrizations are rotations of the top patch parametrization.Source: Appendix B · p. 28
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